Uncertainty Propagation Calculator

Combined uncertainty

What is uncertainty propagation?

Uncertainty propagation estimates the uncertainty in a calculated result when the input measurements each contain uncertainty.

The propagation rule depends on whether the measurements are added, subtracted, multiplied, or divided.

Addition and subtraction

Add absolute uncertainties

This rule applies to both addition and subtraction because the uncertainty widths combine regardless of the operation sign.

Multiplication and division

Add relative uncertainties

After combining the relative uncertainties, multiply by the absolute result to obtain the propagated absolute uncertainty.

Worked example

Multiply 10 ± 0.2 by 5 ± 0.1

1. Calculated value: 10 × 5 = 50.
2. First relative uncertainty: 0.2 ÷ 10 = 0.02.
3. Second relative uncertainty: 0.1 ÷ 5 = 0.02.
4. Combined relative uncertainty = 0.04.
5. Propagated absolute uncertainty: 50 × 0.04 = 2.
6. Final result = 50 ± 2.

Zero results

Relative uncertainty can be undefined

Addition or subtraction can produce a result of zero while still having a valid absolute uncertainty. In that case, relative and percentage uncertainty are undefined because division by zero is not possible.

Scope

Understand the propagation model

• The calculator uses standard classroom worst-case propagation rules.
• It assumes the input uncertainties are independent estimates.
• More advanced statistical analysis may use quadrature or covariance-based methods.
• The final uncertainty should be rounded consistently with the reported value.

Related resources

Build a complete uncertainty workflow

Start with the Measurement Uncertainty Calculator to convert a single measurement into relative and percentage uncertainty.

Then review the Significant Figures Guide and the Lab Report Discussion Guide for reporting and interpretation.

Questions and answers

Uncertainty propagation FAQ